The circle of fifths just blew my mind

In tandem with violin practice, I’m working my way through Practical Theory Complete: A Self-Instruction Music Theory Course. It starts out REALLY basic, with simple notation and rhythms, but works all the way up to composing your own song (!). I just hit lesson 39 (of 86) and my brain exploded.

I’d heard about the “circle of fifths” before, but had never delved into what it actually meant. What it provides is a nifty arrangement of the various (major) keys, anchored by the key of C, that reveals patterns in the progression of sharps and flats that comprise each key’s signature. Check out this awesome magic:

Starting from the key of C, if you go up a fifth, you reach G. The key of G introduces one sharp, F#. Up another fifth from G, you get D, which in addition to F# also features C#. And so on. (The order of keys G-D-A-E are easy to remember for violin players, since those are the four fifth-separated strings on the instrument.) Going down from C a fifth, you get F. The key of F introduces one flat, B♭. Down another fifth is the key of B♭, which adds E♭. And so on.

This defines a linear relationship between C and the keys “above” it as well as “below”; but positioning them on a circle reveals a bit more of the magic: three of these keys are redundant (or “enharmonic”: they sound the same but are notated differently). This diagram shows that G♭ and F# are the same key; my workbook’s diagram also shows that D♭ and C# are enharmonic, as are C♭ and B. And hey, look on any keyboard and what do you see? These key pairs are, in fact, literally the same key.

Want more magic? What’s going on here is modular arithmetic! Not mod 7, but mod 13: the set of values includes { C, C#, D, D#, E, F, F#, G, G#, A, A#, B, B#, C }. For each key, the major scale is traditionally given as WWHWWWH, where W = “whole step” and H = “half step”. But let’s instead view a scale starting on x as the sequence

{ x, x+2, x+4, x+5, x+7, x+9, x+11, x+12 }.

So the key of C contains { C, D, E, F, G, A, B, C }; C+12 = C in this modular land. Now if we go up a fifth and examine the key of G, that’s the same as adding 5 to all entries. The key of G is therefore
{ C+5, D+5, E+5, F+5, G+5, A+5, B+5, C+5 } which yields
{ G, A, B, C, D, E, F#, G } after doing the addition mod 13.

That is, it’s as if we jumped 5 items forward, but then the extra whole-step in the second tetrachord threw off the pattern and caused the F to become an F#. If you move on to the key of D, the first four notes again are unchanged (with respect to the key of G): { D, E, F#, G }, but then we have to shift one note in the second tetrachord again, yielding { A, B, C#, D }. In this way, the sharps keep building on themselves, and the new sharps introduced in each key alternate. The sharp order is F#, C#, G#, D#, A#, E#, B# (see the pattern?). A similar process explains the progression of flats going “down” from C.

This relationship seems also to explain the conventional structure in how key signatures are written. The key of B major has five sharps, which are C#, D#, F#, G#, A# if you write them in ascending order, but F#, C#, G#, D#, A# if you write them in this circle-of-fifths-inspired order. And that seems to be just what one does (see right).

Patterns! Math! Music! And of course, at the heart of this magic is… physics. :)

Ingenuity Engines

I’m John Lienhard, at the University of Houston, where we’re interested in the way inventive minds work.

Well, I’m not John Lienhard, but I’m delighted to make his acquaintance. I just discovered the excellent podcast, “Engines of Our Ingenuity”, which “tells the story of how our culture is formed by human creativity.” Each ~4-minute episode covers some interesting concept, device, invention, inventor, or tidbit of history that bears on our world today. The first one that came up when I subscribed was on The Calculus—and the delightfully accessible and meaningful way in which this subject was covered won me over immediately. Calculus, the mathematics of change, is described as “not any harder than algebra,” just having “quite a different look and feel.”

A beautiful moment in this episode is when Dr. Lienhard connects the concept of calculus to our own lives:

We all see our lives as fleeting moments and as the sum of fragments almost too small to notice.

and then proceeds to quote Joan Didion (Run, River):

“Was there ever in anyone’s life span a point free in time, devoid of memory, a night when choice was any more than the sum of all the choices gone before?”


This show has been going on since 1988. What a wealth of archives to explore! I very much look forward to learning more from Dr. Lienhard about the basis of inventions, and I’m grateful that people out there take the time to create such impressive, useful, fascinating content!

Mathematics and music

My friend Jon Stokes recently posted a delightful mapping from chess moves to music, including several compositions based on famous chess games. This is exactly the kind of geekery that I find most enjoyable, a quest that seeks both interesting new patterns and interesting new ways to experience what we already know. I also learned the word polyrhythm, which occurs when two different rhythms are played against each other. This is frequently found in African music, but apparently occurs even in music by the Beatles (e.g., “Happiness is a Warm Gun”).

He then followed this post by mapping the Fibonacci sequence to music, yielding a lovely fugue and some interesting analysis. I hadn’t known that the Fibonacci sequence modulo 7 produces a repeating sequence — and handily (for 4/4 music), one of length 16! More fun fodder for the future, should I ever be teaching recursion again. I’m charmed by this process of converting an abstraction into the auditory equivalent of a visualization (auditorialization?).

Nice work, Jon!

Multiplication eureka

From the What I Learned Yesterday files…

I have always loved numbers, especially in terms of manipulating them. Remember those arithmetic drill books, endless columns of 3 + 6 and 12 – 5 and every other possible combination? I loved paging my way through them, filling up all of the blank spots. My grandmother’s living room had a big bay window with a flat base I could crawl onto (I don’t think it was intended as a seat), behind the curtains, and I loved to hole up there with that book. Even *better* was when I encountered those drills in elementary school, and they were *timed*! Hooray, a race!

But even before those memories glows a beautiful eureka moment I hope I’ll never forget. I was in daycare, somewhere between 4 and 5 years old, musing about multiplication (for no reason I can recall), when suddenly I Got It. I jumped up and ran around trying to share the shining vision that I’d had. The best I could do then was, “But it’s so simple! Two times two is just 2, two times!”

I still remember those words, and I remember the lack of a similarly excited response from the other kids. Was it incomprehension? Disinterest? I couldn’t seem to put my revelation into words that made sense to anyone else, and I was buzzing with commingled frustration and joy. At that moment, the “x” sign had ceased being an arbitrary symbol specifying a relation to be memorized. Instead, it had *meaning*. I was swimming in triumph at the feeling of having *cracked the code*, seeing yet another pattern but also the whys behind the pattern. (Of course, when I reached elementary school, I then got to memorize the multiplication tables, like everyone else. So much for eureka…)

I have a handful of other memories from that daycare. Conspiring with a friend to stash our pears from lunch, which we hated, in our pockets for later disposal. Sprawling on threadbare green carpet in front of the TV and goggling at afternoon cartoons. Singing “This Old Man, he played one, he played knick-knack on his thumb…” Discovering awe and predation on finding a black widow spider out back. Discovering how surprisingly hard other kids can pinch if you don’t wear green on St. Patrick’s Day. Shivering at horror stories about loose baby teeth being tied to a door and extracted with a slam, then rushing to the bathroom to inspect my own teeth for any worrisome looseness. But these have all faded in a way that my “2, two times” moment has not. And it left me with an appetite for that feeling of “Oh wow, I get it!” that is what makes the study of anything new so very delicious. More learndorphins, anyone?